Question

if$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{,}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$find the following partial derivatives.

${\left(\frac{\partial z}{\partial x}\right)}_{\mathbf{y}}$.

Short answer

The value of provided equation is ${\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{y}}=2x$.

Step-by-step solution

Explanation of solution

In this question it is provided that $z=x^2+2y^2;x=r\cos \theta ,y=r\sin \theta$and to find ${\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{y}}$ .

Partial differentiation

The procedure of calculating the partial derivative of a function is called partial differentiation. This method is used to get the partial derivative of a function with respect to one variable while keeping the other constant.

Calculation

Consider that,

$z=x^2+2y^2$

Taking the partial derivative of z with respect to x ,

$$\begin{gathered} {\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{y}}=\frac{\partial }{\partial \mathrm{x}}={\overline{)\left(x^2+2y^2\right)}}_y \\ {\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{y}}=2x \end{gathered}$$

Hence the value of provided equation is ${\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{y}}=2x$.